On Euler and Fibonacci Num- bers Why Pi is Bounded by Twice Phi G. K. van der Wal On Euler and Fibonacci Numbers Why Pi is Bounded by Twice Phi by G. K. van der Wal to obtain the degree of Bachelor of Science at Delft University of Technology, Department of Applied Mathematics. Student number: 4602501 Project duration: May 1, 2020 – September 11, 2020 Thesis committee: R. Fokkink, TU Delft, supervisor Y. van Gennip, TU Delft Preface This is the report “On Euler and Fibonacci Numbers: Why Pi is Bounded by Twice Phi", which is based on the article from Alejandro H. Morales, Igor Pak & Greta Panova [1], where a link is made between the Fibonacci and Euler numbers. It is meant for mathematicians without the prior knowledge that is needed to understand this article. This report has been written to obtain the degree of Bachelor of Science at Delft University of Technology, Department of Applied Mathematics. It has been written in the period from May 2020 to September 2020 at Delft University of Technology under the supervision of Robbert Fokkink. I would like to thank Robbert Fokkink for his supervision and support. G. K. van der Wal Delft, September 4, 2020 i Abstract In this report, we will look at the connection between the Fibonacci and Euler numbers. By using a combina- torial argument including the Fibonacci and Euler numbers, we will prove our main theorem: F E n! n ¢ n ¸ From the main theorem and the asymptotics of these numbers, we will conclude that ¼ 2'. We follow the · proof in the article of Alejandro H. Morales, Igor Pak & Greta Panova [1], but we will give a more detailed proof and some extra facts about the Golden Ratio, ', and the Fibonacci and Euler numbers. Finally, the article discusses the number of linear extensions of certain partially ordered sets, or posets. We see that there exist a two-dimensional poset Un and complement poset U n, both with n elements, such that the number of linear extensions are respectively En and Fn. We conclude that the Fibonacci and Euler numbers are related to each other. ii Contents Abstract i 1 The Golden ratio 1 1.1 Definition.............................................1 1.1.1 ' as Continued Fraction and Square Root.........................2 1.2 The Golden Ratio in Architecture, Art, the Human Body and Nature...............3 1.2.1 The Golden Spiral.....................................4 1.2.2 The Golden Angle.....................................6 1.3 The Golden Ratio in Geometry..................................7 2 The Fibonacci Numbers9 2.1 Properties of the Fibonacci Numbers...............................9 2.1.1 Binet’s Formula...................................... 10 2.1.2 Fibonacci numbers in Nature............................... 12 2.1.3 The Generating Function of the Fibonacci Numbers.................... 14 2.1.4 Asymptotics of the Fibonacci Numbers.......................... 14 2.2 Bn ................................................ 16 2.3 Extra Facts About the Fibonacci Numbers............................. 17 3 The Euler Numbers 18 3.1 Alternating Permutations..................................... 18 3.2 Properties of the Euler Numbers................................. 19 3.2.1 The Seidel-Entringer Triangle............................... 19 3.2.2 Generating Function of the Euler Numbers........................ 21 3.2.3 Asymptotics of the Euler Numbers............................. 25 4 Combining the Fibonacci and Euler numbers 26 4.1 Combinatorial Proof of the Main Theorem............................ 26 4.2 Combining the Asymptotics of the Fibonacci and Euler Numbers to prove that ¼ 2' ...... 28 · 4.3 Linear Extensions of Partially Ordered Sets............................ 28 Bibliography 34 iii 1 The Golden ratio 1.1. Definition Euclid of Alexandria, sometimes called “the founder of geometry" [2], was the first to give a clear definition of the Golden Ratio, around 300 B.C. Euclid defined the ratio as follows: “a straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser."[3] Meaning: in Figure 1.1, the ratio of the length of the whole line, a b, to that of the greater segment, a, is Å the same as the ratio of the length of the greater segment, a, to the lesser segment, b, i.e., a b a Å a Æ b Figure 1.1: Line segments in the Golden Ratio, where a b is to a as a is to b.[4] Å Using this definition, we can calculate what this ratio is. a a b Å b Æ a a b 1 b Æ Å a a Let x Æ b 1 x 1 Æ Å x x2 x 1 (1.1) Æ Å x2 x 1 0 ¡ ¡ Æ 1 p5 x § Æ 2 The ratio of the line segments should be a positive number, therefore, we define the Golden Ratio as follows: Definition 1.1.1. The Golden Ratio is the positive solution to x2 x 1 0 (equation 1.1), which we denote 1 p5 ¡ ¡ Æ ' Å 1,618034. Æ 2 ¼ 1 1.1. Definition 2 1 p5 We denote the negative solution to equation 1.1 by à ¡ 0,618034. à is sometimes called the little Æ 2 ¼ golden number and it can be expressed in terms of ': 1 p5 1 p5 ' à Š¡ 1 Å Æ 2 Å 2 Æ 1 à 1 ' Æ ¡ Æ¡' There also exists a silver ratio, ±S . If we look at Figure 1.1, a and b are in the silver ratio if the ratio of the length of the greater segment, a, to the lesser segment, b, is the same as the ratio of the sum of the smaller segment plus twice the greater segment, 2a b, to that of the greater segment, a, i.e., Å 2a b a Å a Æ b If we calculate this in the same way as in equation 1.1, we find that the silver ratio is the positive solution to the equation x2 2x 1 0, which we denote ± 1 p2. ¡ ¡ Æ S Æ Å Euclid defined the Golden Ratio, but he named it “extreme and mean ratio" [3]. It was not called the Golden Ratio until 1835, when it was first used by Martin Ohm in the book Die reine Elementar-Mathematik. He named the division of a line AB at a point C such that AB CB AC 2 the ‘golden section’ (or goldener ¢ Æ Schnitt in German) [6]. In Figure 1.2 we have such a line. Figure 1.2: Line AB divided at point C such that AB CB AC 2. ¢ Æ If we rewrite its property, we get: AB CB AC 2 ¢AB Æ AC AC Æ CB If we let the length of AC be a and the length of CB be b, we have: a b a Å a Æ b Which is the same as Euclid’s definition. 1.1.1. ' as Continued Fraction and Square Root We define the recursive sequence (x ) by x 1 1 for all n 1 and x 1. We get: n n 1 n xn 1 0 ¸ Æ Å ¡ ¸ Æ x 1 0 Æ 1 x1 1 Æ Å 1 1 x2 1 Æ Å 1 1 Å 1 1 x3 1 Æ Å 1 1 1 1 Å Å 1 . If we let n go to infinity, we get a continued fraction. We call this expression x. 1 lim xn 1 x n Æ Å 1 Æ !1 1 1 ... Å Å 1.2. The Golden Ratio in Architecture, Art, the Human Body and Nature3 We can then see the same expression in the denominator of the fraction: 1 x 1 Æ Å 1 1 Å 1 ... 1 Å x 1 Æ Å x This is equation 1.1 and x is a positive number, so the solution is: lim xn x ' n !1 Æ Æ 1 ' 1 (1.2) Æ Å 1 1 1 ... Å Å This only holds under the assumption that the sequence converges to x. In section 2.1, we simplify the frac- tions x1, x2, x3, etc. and we will see that they equal the ratio between two consecutive Fibonacci numbers. Then, we prove in Proposition 2.1.2 that these fractions do converge, so equation 1.2 is indeed well-defined. p We can do something similar with square roots. We define the recursive sequence (yn)n 1 by yn 1 yn 1 ¸ Æ Å ¡ for all n 1 and y 1. We get: ¸ 0 Æ y 1 0 Æ y1 p1 1 Æ q Å y 1 p1 1 2 Æ Å Å r q y 1 1 p1 1 3 Æ Å Å Å . If we let n go to infinity, we get an infintely long square root. We call this expression y. r q lim yn 1 1 p1 ... y n !1 Æ Å Å Å Æ Now, we square both sides: r q y 1 1 p1 ... Æ Å Å Å r q y2 1 1 1 p1 ... Æ Å Å Å Å But then we have y again on the right side: y2 y 1 Æ Å This is again equation 1.1 and y is a positive number, so the solution is: lim yn y ' n !1 Æ Æ r q ' 1 1 p1 ... (1.3) Æ Å Å Å 1.2. The Golden Ratio in Architecture, Art, the Human Body and Nature Many people have tried finding this “perfect proportion" in objects such as art, architecture and nature. There has been suggested that the Great Pyramid of Giza and the Parthenon in Athens contain the Golden ratio in their dimensions [5]. But this seems to be untrue, as the measurements of these buildings vary from source to source, so one could easily choose the measurements that would get a result close to '.